Consider a discrete random variable with the following probability mass function. 4-x x = 1,2, 3 f(x) = { , 0, else Compute E[6X – 6 X²].
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- Consider a random variable X taking the values k1, k2, , km E R .. with probability ; Pn E [0, 1] 1. Write down the formula for the P1, P2, .. respectively, where p1 + P2 + expected value of f(X) for a given function f(-). + Pn ...Suppose (S, P) is a binomial distribution with n trials and probability of success p. Let X be the random variable X(k) = k³, where k is the number of successes. Calculate E[X].B5. Let X₁, X₂, ..., Xn be IID random variable with common expectation µ and common variance o², and let X = (X₁ + + X₂)/n be the mean of these random variables. We will be considering the random variable S² given by (a) By writing or otherwise, show that S² (b) Hence or otherwise, show that n S² = (x₁ - x)². = Ĺ(X₂ i=1 X₁ X = (X₁-μ) - (x-μ) = Σ(X; -μ)² - n(X - μ)². i=1 ES² = (n-1)0². You may use facts about X from the notes provided you state them clearly. (You may find it helpful to recognise some expectations as definitional formulas for variances, where appropriate.) (c) At the beginning of this module, we defined the sample variance of the values x₁, x2,...,xn to be S = 1 n-1 n i=1 ((x₁ - x)². Explain one reason why we might consider it appropriate to use 1/(n-1) as the factor at the beginning of this expression, rather than simply 1/n. B6. (New) Roughly how many times should I toss a coin for there to be a 95% chance that between 49% and 510/ of my nain toon land Honda?
- İf X follows the Poisson probability law such that P (xr 1) = P (x = 2). then ind the probability of 4 successes.İf X follows the Poisson probability law such that P (xr 1) = P (x = 2). then ind the probability of 4 successes.3. Determine the value c so that each of the following functions can serve as probability distributions of the discrete random variable X: a. f(x) = c(x² +4) for x = 0,1, 2, 3; 3 b. f(x)%3Dс for x= 0,1, 2 3-x
- The possible values of a discrete random variable X are 0, 1, 3, and 6 with respective probabilities 0.2, 0.3, 0.1, 0.4. Find E[X] and Var(X).A committe of size 3 to be selected at random from 4 chemists and 5 physicists , Find the probability that : E (x)=? а- 16/9 b- 15/9 с-3/4 d- 8/7 e- 4/36. Suppose that the random variables X and Y have joint probability density function given by x+y, 0Let X be a random variable with the probability mass function (PMF) 0.05, =-2 0.25, г — 0 = 1 p(x) = 0.25, г — 2 %3| 0.35, x = 0.1, * = 5 otherwise 0, We can also summarize the key information of the above PME into a table; P(X = =) -2 0.05 0. 0.25 1. 0.25 0.35 0.1 What is the probability that X is non-negative and less than 2? O 0.5 O 0.1 O 0.25 O 0.85 2.Suppose X is a discrete random variable which only takes on positive integer values. For the cumulative distribution function associated to X the following values are known: F(23) 0.34 F(29) = =0.38 F(34) 0.42 F(39) 0.47 F(44) = 0.52 F(49) 0.55 F(56) = 0.61 = Determine Pr[29İf X follows the Poisson probability law such that P (xr 1) = P (x = 2). then ind the probability of 4 successes.Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON